A framework for fitting sparse data

TL;DR

This paper introduces a framework for fitting functions to sparse data by balancing variation (Lipschitz Bound) and deviation, providing deterministic error bounds and applying it to air pollution data.

Contribution

It develops a novel LB-BD trade-off framework with convex optimization methods for fitting sparse data, extending traditional Lipschitz-based approaches.

Findings

The LB-BD function is non-increasing and convex.

Optimal fits with error bounds are derived for given LB and BD.

Application to air pollution data demonstrates practical utility.

Abstract

This paper develops a framework for fitting functions with domains in the Euclidean space, when data are sparse but a slow variation allows for a useful fit. We measure the variation by Lipschitz Bound (LB). Functions which admit smaller LB are considered to vary more slowly. Since most functions in practice are wiggly and do not admit a small LB, we extend this framework by approximating a wiggly function, f, by ones which admit a smaller LB and do not deviate from f by more than a specified Bound Deviation (BD). In fact for any positive LB, one can find such a BD, thus defining a trade-off function (LB-BD function) between the variation measure (LB) and the deviation measure (BD). We show that the LB-BD function satisfies nice properties: it is non-increasing and convex. We also present a method to obtain it using convex optimization. For a function with given LB and BD, we find the optimal fit and present deterministic bounds for the prediction error of various methods. Given the LB-BD function, we discuss picking an appropriate LB-BD pair for fitting and calculating the prediction errors. The developed methods can naturally accommodate an extra assumption of periodicity to obtain better prediction errors. Finally we present the application of this framework to air pollution data with sparse observations over time.